# Hall Effect Measurements Commonly Use Two Sample Geometries Biology Essay

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Electrical characterization of materials evolved in three levels of understanding. In the early 1800s, the resistance R and conductance G were treated as measurable physical quantities obtainable from two-terminal I-V measurements (i.e., current I, voltage V). Later, it became obvious that the resistance alone was not comprehensive enough since different sample shapes gave different resistance values. This led to the understanding (second level) that an intrinsic material property like resistivity (or conductivity) is required that is not influenced by the particular geometry of the sample. For the first time, this allowed scientists to quantify the current-carrying capability of the material and carry out meaningful comparisons between different samples. Theories of electrical conduction were constructed with varying degrees of success, but until the advent of quantum mechanics, no generally acceptable solution to the problem of electrical transport was developed. This led to the definitions of carrier density n and mobility µ (third level of understanding) which are capable of dealing with even the most complex electrical measurements today.

Hall effect measurements commonly use two sample geometries: (1) long, narrow Hall bar geometries and (2) nearly square or circular van der Pauw geometries. Each has advantages and disadvantages. In both types of samples, a Hall voltage is developed perpendicular to a current and an applied magnetic flux. The following is an introduction to the Hall effect and its use in materials characterization

## Hall Bar

Some common Hall bar geometries are shown in Figure. The Hall voltage developed across an 8-contact Hall bar sample with contacts numbered as in Figure is:

where V24 is the voltage measured between the opposing contacts numbered 2 and 4, RH is the Hall coefficient of the material, B is the applied magnetic flux density, I is the current, and t is the thickness of the sample (in the direction parallel to B). This section assumes SI units. For a given material, increase the Hall voltage by increasing B and I and by decreasing sample thickness.

The relationship between the Hall coefficient and the type and density of charge carriers can be complex, but useful insight can be developed by examining the limit B âˆž when:

where r is the Hall scattering factor, q is the fundamental electric charge, p is the density of positive and n the density of negative charge carriers in the material. For the case of a material with one dominant carrier, the Hall coefficient is inversely proportional to the carrier density. The measurement implication is that the greater the density of dominant charge carriers, the smaller the Hall coefficient and the smaller the Hall voltage which must be measured. The scattering factor r depends on the scattering mechanisms in the material and typically lies between 1 and 2.1,

Another quantity frequently of interest is the carrier mobility, defined as:

where is the Hall mobility and Ï € is the electrical resistivity at zero magnetic flux density. The electrical resistivity can be measured by applying a current between contacts 5 and 6 of the sample shown in Figure and measuring the voltage between contacts 1 and 3, then using the formula:

where w is the width and t is the thickness of the Hall bar, b is the distance between contacts 1-3, and B is the magnetic flux density at which the measurement is taken. The Hall bar is a good geometry for making resistance measurements since about half of the voltage applied across the sample appears between the voltage measurement contacts. For this reason, Hall bars of similar geometries are commonly used when measuring magnetoresistance or Hall mobility on samples with low resistances.

## Disadvantages of Hall bar geometries:

A minimum of six contacts to make mobility measurements; accuracy of resistivity measurements is sensitive to the geometry of the sample; Hall bar width and the distance between the side contacts can be especially difficult to measure accurately. The accuracy can be increased by making contact to the sides of the bar at the end of extended arms. Creating such patterns can be difficult and can result in fragile samples.

Figure Common Hall Bar Geometries. Sample thickness, t, of a thin film sample = diffusion depth or layer thickness. Contacts are black, numbered according to the standard to mount in Lake Shore sample holders.

## Hall Effect is useful in determining the properties of semiconductor:

The Hall effect can be achieved by inducing a magnetic field perpendicular to the current flow direction in a semiconductor. Under such conditions, a voltage is developed perpendicular to both the current and magnetic field. This voltage is known as the Hall voltage. The origin of the Hall voltage can be seen by considering the forces on a charged carrier in the presence of a magnetic field (see figure 1):

(1)

The first term is due to the total electric field driving the current through the sample. The second term is due to the Lorentz force on the charged carriers, and tends to deflect the carrier toward the side of the sample. The direction of the deflection depends on the sign of the carrier's charge.

The Hall effect device. Current flows in the positive x-direction. The applied magnetic field is in the positive z-direction. For a p-type sample an internal electric field develops in the positive y-direction.

Consider the example illustrated in Figure 1. Let's assume that we have a p-type semiconductor bar. The applied electric field and the current are in the positive x-direction, the applied magnetic field is in the positive z-direction. The y-component of the force is:

This equation implies that unless something happens, all carriers moving in the sample will experience a force that will drive them toward one side of the sample. In this case, the holes would move in the negative y-direction.

If a number of holes were to collect at the right side of the sample, that side would take on a positive charge relative to the left side. This sets up an internal electric field in the +y-direction. Note that the only applied electric field is in the +x-direction. The force due to the internal electric field opposes the Lorentz force. To maintain a steady flow of current through the sample, we must have a balance of forces:

resulting in no net force on the carriers in the y-direction. The internal field can be set up by moving the holes only slightly to the right.

The presence of the internal field can be detected by measuring the voltage developed across the sample:

where w is the width of the sample. This is known as the Hall voltage.

Carriers subject to an electric field move with a velocity called the drift velocity. The hole current in our sample can be written as

where +q is the hole charge, p is the hole density in /cm3, vd is the drift velocity, and A is the cross sectional area of the sample. If we convert this to an equation for the current density vector, where the magnitude J = I/A and the direction is parallel to the drift velocity, we have

The drift velocity is related to the electric field driving it through a proportionality constant known as the mobility:

Substituting this into the current density equation, we get

Using this relationship in our equation for the field Ey, we get:

where RH =1/qp is called the Hall coefficient.

RH = -1/qn for n-doped samples.

We can also extend this model to consider the Hall effect when both electrons and holes are present, resulting in the following equation (for small fields):

## Applications

1) Doping concentration

Equation (9) can be rearranged as

(11)

We see that we can use measurements of the Hall voltage, magnetic field, current, and sample thickness to determine the Hall coefficient for any sample. From the Hall coefficient we can derive the doping density, p or n. This measurement is a diagnostic tool for determining the doping level in the sample.

2) Mobility

If a measurement of sample resistance R is made, you can calculate the resistivity

(12)

Since the conductivity Ïƒ = 1/Ï is equal to qµpp, the mobility µp is just the ratio of the Hall coefficient and the resistivity. Measurements of the Hall coefficient and the resistivity over a range of temperatures yield plots of majority carrier concentration and mobility vs. temperature, very useful data to have for semiconductors.

3) Current measurement

Another real-world application of Hall effect devices is as a sensor for current measurement. A current (dc or ac) passing through a wire (figure 2) generates a magnetic field:

(13)

where Iw is the current flowing in the wire, and r is the radial distance from the wire. If a Hall device is placed near the wire and a constant current Is is passed through it, the magnetic field generated by the wire will induce a Hall voltage Vy in the device.

Figure 2. End view of a current carrying wire. Current is flowing into the page, so using the right-hand-rule, the magnitic field flux circles the wire in a clockwise sense.

Solving equation 11 for B, we see

(14)

Setting equations 13 and 14 equal and solving for Iw,

(15)

Thus, if we know the details of the sensor and the distance of the sensor from the wire, and measure the Hall voltage we can make a "non invasive" determination of the current flowing in the wire.

The Hall Coefficient measurements provide the following information about the solid

1. The sign (electrons and holes) of charge carriers.

2. The type of material.

3. The carrier concentration can be measured.

4. The mobility of charge carrier.

5. It can be used to determine the given material is insulator or semiconductor.

Background on RTD's:

RTDs is considered among the fastest devices because tunneling is very fast and is not transit-time limited as in CMOS technology, etc. RTDs provide a low leakage current when a reverse bias is applied. Large dynamic range within a small input voltage range However, the output current and power of RTDs is very limited compared to CMOS. RTDs is much faster than any other conventional transistor. Very important alternative as transistor technology continues to scale down to the nanometer range Very good rectifier - low leakage current Much research needs to be done to improve the output power and also to integrate them with conventional transistors

## Need for RTD:

Today' s modern era of information technology is due to high speed compact and low cost the electronic representation and processing of information. For continuation growth of this, it demands further reduction of chip size. Chip size has been following the moore's law for last three decades and it seems continue to apply for some time in future. Eventually the downscaling of conventional transistors and integrated circuits (IC's) will eventually be reached . While the downscaling of conventional transistors enjoys an exceptional, rapid evolution, revolutionary device concepts have been actively sought, particularly in the two related areas known as nanoelectronics and single electronics. The RTD, and its several variations, has become a research focus in nanoelectronics for its promise as a primary nanoelectronic device for both analog and digital applications.

## Why RTD:

It is well known that when the size of a system becomes comparable to the electron wavelength, quantum effects become dominant. This occurs when transistors are

downscaled and their characteristic dimensions reach the nanometer range, leading to new phenomena and possible novel devices based on quantum tunneling mechanisms. For nanoelectronics to become a reality, it is essential that the new devices and circuits must be fabricated with nanometer precision, and one must be able accurately to design the devices and circuits

This temperature requirement is the single most important feature that any new technology must satisfy. It is what distinguishes the RTD from other interesting quantum device concepts that have been proposed but that show weak, if

any, desired phenomena at room temperature

RTD is promising candidate for digital circuit applications due to its negative differential resistance (NDR) characteristic, structural simplicity, relative ease of fabrication, inherent high speed, flexible design freedom, and versatile circuit functionality. There is a good practical reason to believe that RTD's may be the next device based on quantum confined heterostructures to make the transition from the world of research into practical application. Progress in epitaxial growth has improved the peak-to-valley current ratio at room temperature even beyond that required for many circuit applications.

RTD applications:

The main issue at present is not, in fact, the RTD performance itself but the monolithic integration of RTD's with transistors [high electron mobility transistors (HEMT's) or heterojunction bipolar transistors (HBT's)] into integrated circuits with useful numbers and density of devices

## Features of RTD:

Resonant tunnelling:

Resonant tunneling refers to tunneling in which the electron transmission coefficient through a structure is sharply peaked about certain energies. A resonant tunnelling diode (RTD) typically consists of an undoped quantum well layer sandwiched between undoped barrier layers and heavily doped emitter and collector contact regions

The basic RTD device configuration is a DBQW structure of nanometer dimensions, including two contacts as depicted in Fig. 1, where the regions I, II and VI, VII are heavily doped contacts made from a semiconductor with a relatively small bandgap. These layers comprise the emitter and collector, respectively. Regions III and V are quantum barriers made from a semiconductor with a relatively larger bandgap. Region IV between the two barriers is the quantum well made again from

the smaller bandgap semiconductor. It is sometimes also called the base

Envision a spectrum of electrons in region I, driven by a bias voltage applied across the RTD contacts, incident upon the DBQW structure

P-N diode with heavy doping (1020 cm-3) in both regions (Degenerately doped). The depletion region is very narrow (<10nm). High concentration of electrons in the conduction band of N-type and holes in the valence band of P-type material

Apply increasing forward bias voltage Starting at zero bias

Electrons in N-region conduction band are energetically aligned to the holes in the valence band of P-region. Tunneling occurs. Forward current is produced.

As you increase the bias voltage, a maximum current will be produced when all electrons are aligned with the holes

As bias voltages continues to increase, current will decrease because less electrons are aligned with the holes

As the bias voltage continues to increase, electrons are no longer energetically aligned with the holes and the diffusion current dominates over tunnelling .Reverse bias voltage is very low breakdown. and have high leakage current so its not a good rectifier

Electrons must have a certain minimum energy above the energy level of the quantized states in the quantum well in order for tunneling to occur. Once the bias voltage is big enough to provide enough energy, RTDs looks like a normal TD in reverse bias, RTDs do not have large leakage current

## NDR:

Characterized by the current peak to valley ratio (PVR=I/V).To achieve maximize dynamic range, high PVR is desired. And to obtain maximum output power from RTD, high current density is required. Decrease the thickness of the quantum well barrier. Increase emitter doping level.However, PVR will be decreased and leakage will increase

NEW

## RESONANT TUNNEL EFFECT

Electrons in heterojunctions and in quantum wells can respond with very high mobility to applied electric fields parallel to the interfaces. Under certain circumstances, electrons can tunnel through these potential barriers, constituting the so-called perpendicular transport. Tunnelling currents through heterostructures can show zones of negative differential resistance (NDR), which arise when the current level decreases for increasing voltage.

The NDR effect was first observed by Esaki when studying p-n junction tunnel

diodes in 1957 and, together with Tsu, proposed in the 1970s that this effect would be also observed in the current through quantum wells. However, it was not until the mid 1980s that the experimental growth deposition systems for heterostructures allowed the standard fabrication of quantum well devices based on NDR effects.

The operation of NDR quantum well electronic devices is based on the so-called

resonant tunnel effect (RTE), which takes place when the current travels through a structure formed by two thin barriers with a quantum well between them. The I-V characteristics of RTE devices are somewhat similar to that of Esaki's tunnel diode.

Figure (a) shows the representation of the conduction band of a double heterojunction with a quantum well between the junctions. The thickness of the quantum well is supposed to be small enough (5-10 nm) as to have only one allowed electron energy level E1 (resonant level). The well region is made from lightly doped GaAs surrounded by higher gap AlGaAs. The outer layers are made from heavily doped n-type GaAs (n+ GaAs) to facilitate the electrical contacts. The Fermi level of the n+ GaAs is represented within the conduction band, since it can be considered a degenerated semiconductor

Schematic representation of the conduction band of a resonant tunnel diode: (a) with

no voltage applied; (b), (c), and (d) for increasing applied voltages; (e) current-voltage

characteristic

Suppose that an external voltage, V, is applied, starting from 0V. It can be expected that some electrons tunnel from the n+ GaAs conduction band through the potential barrier, thus resulting in increasing current for increasing voltage (region 1-2 in the I-V curve near 0 V). When the voltage increases, the electron energy in n+ GaAs increases until the value 2E1/e is reached, for which the energy of the electrons located in the neighbourhood of the Fermi level coincides with that of level E1 of the electrons in the well (Figure (b)). In this case, resonance occurs and the coefficient of quantum transmission through the barriers rises very sharply. In effect, when the resonant condition is reached, the electron wave corresponding to the electrons in the well is coherently reflected between the two barriers (this is analogous to the optical effect produced in Fabry-Perot resonators). In this case, the electron wave incident from the left excites the resonant level of the electron in the well, thus increasing the transmission coefficient (and thus the current) through the potential barrier (region 2 in the I-V characteristic).

In this condition, the effect is comparable to electrons impinging from the left being

captured in the well and liberated through the second barrier. If the voltage is further

increased (Figure (c)), the resonant energy level of the well is located below the

cathode lead Fermi level and the current decreases (region 3), thus leading to the socalled

negative differential resistance (NDR) region (region 2-3). Finally, for even higher

applied voltages, Figure(d), the current again rises due to thermo-ionic emission over

the barrier (region 4).

Commercial resonant tunnelling diodes (RTDs) used in microwave applications are

based on this effect. A figure of merit used for RTDs is the peak-to-valley current ratio (PVCR), of their I-V characteristic, given by the ratio between the maximum current (point 2) and the minimum current in the valley (point 3). Although the normal values of the figure of merit are about five for AlGaAs-GaAs structures at room temperature, values up to 10 can be reached in devices fabricated from strained InAs layers, surrounded by AlAs barriers and operating at liquid nitrogen temperature.

If RTDs are simulated by a negative resistance in parallel with a diode capacitance

C and a series resistance RS, as is the case of normal diodes, it is relatively easy to

demonstrate that the maximum operation frequency increases as C decreases. The resonant tunnel diode is fabricated from relatively low-doped semiconductors, which results in wide depletion regions between the barriers and the collector region, and accordingly, small equivalent capacity. For this reason, RTDs can operate at frequencies up to several terahertzes (THz), much higher than those corresponding to Esaki's tunnel diodes which just reach about 100 GHz, with response time under 10âˆ’13 s. Small values of the negative differential resistance, i.e. an abrupt fall after the maximum of the I-V curve result in high cut-off frequencies of operation. In fact, RTDs are the only purely electronic devices that can operate up to frequencies close to 1 THz, the highest of any electron transit time device.

In a general sense, the power delivered from the RTDs to an external load is small and the output impedance is also relatively small. For this reason, it is sometimes hard to adapt them to the output of waveguides or antennas. The output signal is usually of low power (a few milliwatts) since the output voltage is usually lower than 0.3V, due to the values of the barrier heights and energy levels in quantum wells. RTDs have been used to demonstrate circuits for numerous applications including static random access memories (SRAM), pulse generators, multivalued memory, multivalued and self-latching logic, analogue-todigital converters, oscillator elements, shift registers, low-noise amplification, MOBILE logic, frequency multiplication, neural networks, and fuzzy logic. In particular, for logic applications, values of PVCR of 3 or higher and a high value of the peak current density, Jp, are required. In the case of memory applications, the ideal PVCR is 3 and values of Jp of a few Acmâˆ’2 are more appropriate. High frequency oscillators always require high Jp with PVCR over 2..